Intuitionistic Fuzzy Contra Open Mappings in Intuitionistic Fuzzy Topological Space

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014

Intuitionistic Fuzzy Contra Open Mappings in Intuitionistic Fuzzy Topological Space A. MANIMARAN, K. ARUN PRAKASH  Abstract— In this paper, we introduce intuitionistic fuzzy contra open mappings. We study some of their properties and obtain their characterizations. Relationship between this new class and other classes of functions are established in intuitionistic fuzzy topological spaces.

Index Terms— Intuitionistic fuzzy topological spaces, Intuitionistic fuzzy open sets and Intuitionistic fuzzy closed sets, Intuitionistic fuzzy contra open mappings.

Definition 2.2:[1] Let X be a non-empty set and let the IFS’s and in the form A B , . Let {A j : j  J} be an

arbitrary family of IFS’s in X,   . Then, i) if and only if and ii)

AMS Classification: 54A40, 54D20.

[ ]; ;

iii)

;

iv)

;

v) 1  { x, 1, 0  ; x  X } and 0  { x, 0, 1  ; x  X };

I. INTRODUCTION

~

After the introduction of fuzzy sets by Zadeh [6], there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them. Using the notion of intuitionistic fuzzy sets, Coker [2] introduced the notion of intuitionistic fuzzy topological spaces. In this paper, we introduce the concept of intuitionistic fuzzy contra open mapping. We have also studied some of the properties of intuitionistic fuzzy contra open mapping and obtain their characterizations.

vi)

~

where the mappings μA : X  [0,1] and : X  [0,1] denote the degree of the membership (namely μA(x)) and the degree of non-membership (namely νA(x)) of each element x X to the set A respectively, 0  μA (x) + (x) ≤ 1 for each x  X. Obviously, every fuzzy set A on a non-empty set X is an IFS having the form .

~

~

Y are two non-empty sets be

a

B

pre-image of f is denoted and defined by

1

function.

Y,

If

then the under

are fuzzy sets, we explain

B  (x)  B f (x) .

Definition 2.4: An intuitionistic fuzzy topology [2] (IFT, for short) on a non-empty set X is a family  of IFS’s in X satisfying the following axioms: i) 0 , 1   . ~

Definition 2.1: [1] Let X be a nonempty set. An intuitionistic fuzzy set (IFS) A in X is an object having the form

~

is an IFS in

that f Before entering to our work, we recall the following notations, definitions and results of intuitionistic fuzzy sets.

Throughout this paper, , and are always means intuitionistic fuzzy topological spaces in which no separation axioms are assumed unless otherwise mentioned.

A  A, 0  1 and 1  0

Definition 2.3: [2] Let X and f : X,   Y,  and

Since  B ,

II. PRELIMINARIES

~

~

ii) A1  A 2   for some A1 , A 2  . iii)  A j  τ for any {A j : j  J} τ.

In this case, the ordered pair X,  is called an intuitionistic fuzzy topological space (IFTS, for short) and each IFS in  is known as an intuitionistic fuzzy open set (IFOS, for short) in X . The complement of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set (IFCS, for short).

Definition 2.5: [2] Let

X, 

be an IFTS and let

be an IFS in X . Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A is defined by

Manuscript received April 15, 2014.

A. MANIMARAN, Department of Mathematics, Kongu Engineering Colleg, Perundurai-638 052, Erode, Tamilnadu, India.

K. ARUN PRAKASH, Department of Mathematics, Kongu Engineering College, Perundurai-638 052, Erode ,Tamilnadu, India.

int(A)    G / G is an IFOS in X and G  A  cl(A)    K / K is an IFCS in X and A  K  .

160

www.erpublication.org

Intuitionistic Fuzzy Contra Open Mappings in Intuitionistic Fuzzy Topological Space

Remark 2.6:[2]

For any IFS

A

in

X, 

IFCS in . Therefore, mapping. ,we

have cl(A)  int(A), int(A)  cl(A) .

Definition 2.7:[4] Let A be an IFS in an IFTS (X, τ). Then the intuitionistic fuzzy semi - interior and intuitionistic fuzzy semi - closure of A are defined by sint(A) =  {G | G is an IFSOS in X and G  A}, scl(A) =  {K | K is an IFSCS in X and A  K} respectively.

Obviously scl(A) is the smallest IFSCS which contains A and sint(A) is the largest IFSOS which is contained in A. Definition 2.8: Let A be an IFS in an IFTS (X, τ). Then A is called an intuitionistic fuzzy semiopen set (IFSOS) [3] of X if .

The complement of intuitionistic fuzzy semiopen set is called the intuitionistic fuzzy semiclosed set.

Definition 2.9: An IFS A in an IFTS X,   is called an intuitionistic fuzzy regular open set [3] (IFROS) if int (cl(A)) = A. The complement of intuitionistic fuzzy regular open set is called intuitionistic fuzzy regular closed (IFRCS, for short). The family of all IFROS (IFRCS) of X,   is denoted by IFROS X 

(IFRCS X  ).

Definition 2.11: [5] An intuitionistic fuzzy point (IFP for short), written p ,  is defined to be an IFS of X given by

 α, β  p α,β  x     0,1

, Let and

Theorem 3.5: Let be a bijective mapping and IFPC( ) = IFC( ). Suppose that one of the following properties hold: (i) for each IFS in

Definition 2.13: [3] Let A be an IFS in an IFTS (X, τ). Then A is called an intuitionistic fuzzy preclosed set (IFPCS) if .

III. INTUITIONISTIC FUZZY CONTRA OPEN MAPPING

3.2:

:

Let

,

and

. .

Proof: (i)  (ii) Let be an IFCS in . Then is an IFOS in . By hypothesis is an IFCS in .

Since , is an IFOS in . (ii)  (i) Let be an IFOS in . Then is an IFCS in .

By Hypothesis is an IFOS in . Hence is an IFCS in . Thus is an intuitionistic fuzzy contra open mapping.

Theorem 3.4: For a bijective mapping the following statements are equivalent: i) is an intuitionistic fuzzy contra open mapping; ii) for every IFCS in , is an IFOS in ; iii) for every IFCS in and for any IFP , if , then ; iv) For any IFCS in and for any , if , then there exists and IFOS such that and .

is said to be an is an IFCS in

Definition 2.12:[5] An IFP p(,) in X is said to be quasi-coincident with an IFS , denoted by p(,) q A, if and only if or .

Example

Theorem 3.3: Let be a surjective mapping. Then the following statements are equivalent: i) is an intuitionistic fuzzy contra open mapping; ii) is an IFOS in for every IFCS in .

Proof: The proof of (i)  (ii) and (ii)  (i) follows from Theorem 3.3. (ii)  (iii) Let be an IFCS and let , Assume that . Then . By hypothesis is an IFOS in . then .

Hence . (iii)  (ii) Let be an IFCS in and , Assume that . Then . By hypothesis, .

Hence which implies is an IFOS in . (iii)  (iv) Let be an IFCS in and , Assume that . Then . By hypothesis, . Thus is an IFOS in .

Put , then and . (iv)  (iii) Let be an IFCS and such that , By hypothesis there exists an IFOS in such that and . Let . Then and since is an IFOS, is an IFOS which implies .

if x  p otherwise

Definition 3.1: A mapping intuitionistic fuzzy contra open mapping if for every IFOS in .

is an intuitionistic fuzzy contra open

;

Then,

and are IFTs on and respectively. Define a mapping by and . Clearly, , which is an

161

(ii)

for each IFS

in

;

www.erpublication.org

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014 (iii)

.

for each IFS

.

in ;

(iv)

for each IFOS

=

in ; then,

Therefore,

. Hence,

is an IFOS in . By Theorem 3.5, fuzzy contra open mapping.

is an intuitionistic fuzzy contra open mapping.

Proof: (i)  (ii) can be proved by taking the complement in (i) (ii)  (iii) Let be an IFS in . Put in , this implies in .

Now by (ii).

every IFOS

in ; for

(ii) every IFCS

be an IFOS in , then

. By the

hypothesis,

.

Therefore,

.

Suppose that (iv) holds. Let be an IFOS in . then is an IFOS in . Hence by hypothesis,

. . Now,

Therefore,

is an IFPCS in . By

hypothesis is an IFCS in . Hence, fuzzy contra open mapping.

is an intuitionistic

Theorem 3.6: Let be a surjective mapping.

Suppose that one of the following properties hold: (i) for each IFS in ; (ii) for each IFS in ; (iii) Then

is an intuitionistic

Theorem 3.8: Let be a surjective intuitionistic fuzzy contra open mapping, then the following conditions hold: (i) for

Therefore,

. (iii)  (iv) Let

for each IFS

in ;

is an intuitionistic fuzzy contra open mapping.

Proof: (i)  (ii) Let us consider is an IFS in

be an IFS in . Then .

By hypothesis .

Now,

. (ii)  (iii) can be proved by taking the complement in (ii).

Suppose that (iii) holds. Let be an IFCS in . then and is an IFS in . Now, , by hypothesis. This implies is an IFOS in . By Theorem 3.3, f is an intuitionistic fuzzy contra open mapping.

Theorem 3.7: Let be a bijective mapping.

Then f is an intuitionistic fuzzy contra open mapping if for every IFS in .

Proof: Let be an IFCS in . Then an IFS in .

By . Since

and is hypothesis is one to one

162

Now,

Proof: (i) Let hypothesis

in ;

be an IFOS in . Clearly, . By is an IFCS in . This implies that .

(ii) can be proved by taking the complement in (i).

Theorem 3.9: Let be a one-to-one mapping, then the following statements are equivalent: i) is an intuitionistic fuzzy contra open mapping; ii) for each IFP and for each IFCS containing ,there exists an IFOS contained in and such that ; iii) for each IFP and for each IFCS containing ,there exists an IFOS contained in and such that .

Proof: (i)  (ii) Let be an IFCS in and such that .

Then, . By the hypothesis, is an IFOS in . Let . (ii)  (i) Let be an IFCS in and such that . Then . By the hypothesis, is an IFOS in . Let . This implies . (iii)  (i) Let be an IFCS in and . Let . Then by the hypothesis, there exists an IFOS in such that and . This implies .

That is .

Since is an IFOS, .

Therefore, .

But . Hence is an IFOS in . hence is an intuitionistic fuzzy contra open mapping.

Theorem 3.10: A mapping is an intuitionistic fuzzy contra open mapping if for every IFS in .

www.erpublication.org

Intuitionistic Fuzzy Contra Open Mappings in Intuitionistic Fuzzy Topological Space

Proof: Let be an IFCS in . Then . Since every IFCS is an IFSCS, .

By hypothesis . This implies that is an IFOS in . Hence is an intuitionistic fuzzy contra open mapping.

Theorem 3.11: If IFSO(Y) = IFO(Y), then a mapping is an intuitionistic fuzzy contra open mapping if and only if for every IFS

in .

Proof: Necessity: Let be an IFOS in . Then is an IFCS in . By hypothesis, is an IFOS in . Since every IFOS is an IFSOS, is an IFSOS in . Therefore, .

Sufficiency: Let hypothesis

be an IFCS in

. Then

. By .

Then

. Therefore, is an IFSOS in . By hypothesis, is an IFOS in .

Hence, is an intuitionistic fuzzy contra open mapping.

Theorem 3.12: A mapping is an intuitionistic fuzzy contra open mapping if and only if for every IFS in .

Proof: Necessity: Let be an IFS in . Then is an IFCS in .

By hypothesis, is an IFOS in . Then, .

Sufficiency: Let hypothesis,

be an IFCS in . Then . Thus,

. By . Now, is an

intuitionistic fuzzy contra open mapping.

Theorem 3.13: A mapping is an intuitionistic fuzzy contra open mapping, then for every IFS in .

Proof: Let be an IFOS in . Then By hypothesis, is an Then,

is an IFCS in .

IFOS in . ..

Theorem 3.14: An intuitionistic fuzzy open mapping is an intuitionistic fuzzy contra open mapping if IFOS ( ) = IFCS( ).

Proof: Let be an IFOS in . By hypothesis is an IFOS and hence is an IFCS in . Hence is an intuitionistic fuzzy contra open mapping.

Definition 3.15: A mapping is said to be an intuitionistic fuzzy almost contra open mapping, if the image of each IFROS in is an IFCS in .

Theorem 3.16: Every intuitionistic fuzzy contra open mapping is an intuitionistic fuzzy almost contra open mapping.

163

Proof: Let be an intuitionistic fuzzy contra open mapping and be an IFROS in . Since every IFROS is an IFOS, is an IFOS in . By hypothesis is an IFCS in . Hence is an intuitionistic fuzzy almost contra open mapping.

The converse of the above theorem is not true in general as seen from the following example.

Example

3.17:

Let

,

.

Let , and

.

Then

and are IFTs on and respectively. Define a mapping by and . Clearly is an IFROS in and is an IFCS in . Therefore, is an intuitionistic fuzzy almost contra open mapping. Since is an IFOS in , but is not an IFCS in , is not an intuitionistic fuzzy contra open mapping.

Theorem 3.18: Let and be any two mappings, then and the following conditions hold: i) If is an intuitionistic fuzzy open mapping and g is an intuitionistic fuzzy contra open mapping, then is an intuitionistic fuzzy contra open mapping. ii) If is an intuitionistic fuzzy almost contra open mapping and g is an intuitionistic fuzzy contra open mapping, then is an intuitionistic fuzzy almost open mapping. iii) If and are intuitionistic fuzzy contra open mappings, then is an intuitionistic fuzzy open mapping.

Proof: (i) Let be an IFOS in . By the hypothesis is an IFOS in . Since is an intuitionistic fuzzy contra open mapping is an IFCS in . Hence, is an intuitionistic fuzzy contra open mapping. (ii) Let be an IFROS in . By the hypothesis is an IFCS in . Since is an intuitionistic fuzzy contra open mapping is an IFOS in . Hence, is an intuitionistic fuzzy almost open mapping. (iii) Let be an IFOS in . By the hypothesis is an IFCS in . Since is an intuitionistic fuzzy contra open mapping, then by Theorem 3.3, is an IFOS in . Hence, is an intuitionistic fuzzy open mapping.

Definition 3.19: A mapping is said to be an intuitionistic fuzzy contra irresolute open mapping, if is an IFSCS in for every IFSOS in .

Example

3.20:

Let

, . Then,

mapping

.

Let and and

are IFTs on X and Y respectively. Define a by and .

www.erpublication.org

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-2, Issue-4, April 2014

Clearly, every IFSOS in is an IFSCS in . Therefore, an intuitionistic fuzzy contra irresolute open mapping.

is

Theorem 3.21: Let be a surjective mapping, then the following statements are equivalent: i)

is an intuitionistic fuzzy contra irresolute open mapping ii) f is an IFSOS in for every IFSCS in .

Proof: Let be an IFSCS in . Then, is an IFSOS and by the hypothesis is an IFSCS in . Hence, is an IFSOS in .

Let be an IFSOS in . By the hypothesis is an IFSOS in . Then, is an IFSCS in . Hence, is an intuitionistic fuzzy contra irresolute open mapping.

Theorem 3.22: Every intuitionistic fuzzy contra irresolute open mapping is an intuitionistic fuzzy contra open mapping, if IFSC( ) = IFC( ) Proof: Let be an intuitionistic fuzzy contra irresolute open mapping and be an IFOS in X. Since every IFOS is an IFSOS, is an IFSOS in . By hypothesis is an IFSCS in . Since IFSCS( ) = IFCS( ), is an IFCS in . Hence is an intuitionistic fuzzy contra open mapping.

REFERENCES [1] Atanassov K, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87-96.

[2] Coker. D, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems 88 (1997), 81- 89, [3] Gurcay. H, Coker. D, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math. 5 (1997), 365 - 378.

[4] Jun. Y.B, Kong. J.O and Song S.Z, Intuitionistic fuzzy irresolute and continuous mappings, Far East J. Math. Sci. 17(2) (2005), 201-216.

[5] Lee S. J, and Lee E.P, The category of intuitionistic fuzzy topological spaces, Bull. Korean Math. Soc. 37 (2000).

No.63-76.

[6] Zadeh LA. Fuzzy sets, Information and Control, 8 (1965), 338 - 353.

164

www.erpublication.org